College Calculus: Level II Arclength

In this video we are going to talk about Arc Length. There is one main formula for arc length that one need to know and we are going to introduce it. We use formula for arc length when we are trying to find the length of a curve y=f(x) from point a to point b. Where this formula comes from is the Pythagorean Theorem. There is one common mistake that Calculus 2 students make. When they get an arc length problem they usually try to integrate the original function. Doing so, they get an area instead of the arc length. We will try out this formula on several examples.

I think you made a mistake in the first example by saying that u add 1/2 to the function i thought you would -1/2

1 answer

Last reply by: Dr. William MurrayFri Dec 7, 2012 4:48 PM

Post by Riley Argueon June 3, 2012

Excellent lecture.You elegantly and simply explained this, thank you.

3 answers

Last reply by: Dr. William MurrayFri Dec 7, 2012 4:47 PM

Post by Alphonse Mbuon March 7, 2012

why do you use a^2+b^2-2ab. i dont understand what makes that valid

Arclength

Main formula:

Hints and tips:

To
remember this formula, it helps to recall that it comes from the
distance formula between two points, which in turn comes from the
Pythagorean Theorem.

Remember
that you must integrate the square root formula above. A common
mistake is to integrate the function itself, not the square root
formula. Of course, this would give you the area under the curve and
not the arclength.

A
similar mistake is to mix this up with formula for surface area of
revolution, which looks similar. Be careful which one you are asked
for.

Dont
make the common algebraic mistake of thinking that
reduces to a + b! This is extremely wrong, and your
teacher will likely be merciless if you do it!

Many
problems in Calculus II classes are rigged so that when you
expand 1 + f ′(x)²
, it becomes a perfect square that cancels nicely with the square root.

Often
this perfect square is achieved by making the f ′(x)²
be something of the form (a − b)² = a²
− 2ab + b². Then the +1 changes it to a²
+ 2ab + b², which you can then factor as (a +
b)².

When
its feasible, check that your answer makes sense. Unlike area
integrals, which can be negative if a curve goes below the x-axis,
arclength should always be positive! You might also be able to check
that the curve looks about as long as your answer.

Arclength

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.